A note on bounds for the cop number using tree decompositions

In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth

Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in $[0,1/2].

The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

Structure of aluminum atomic chains

First-principles density functional calculations reveal that aluminum can form planar chains in zigzag and ladder structures. The most stable one has equilateral triangular geometry with four nearest neighbors; the other stable zigzag structure has wide bond angle and allows for two nearest neighbors. An intermediary structure has the ladder geometry and is formed by two strands. All these planar geometries are, however, more favored energetically than the linear chain. We found that by going from bulk to a chain the character of bonding changes and acquires directionality. The conductance of zigzag and linear chains is 4e^2/h under ideal ballistic conditions.Comment: modified detailed version, one new structure added, 4 figures, modified figure1, 1 tabl

Structural Transitions and Global Minima of Sodium Chloride Clusters

In recent experiments on sodium chloride clusters structural transitions between nanocrystals with different cuboidal shapes were detected. Here we determine reaction pathways between the low energy isomers of one of these clusters, (NaCl)35Cl-. The key process in these structural transitions is a highly cooperative rearrangement in which two parts of the nanocrystal slip past one another on a {110} plane in a direction. In this way the nanocrystals can plastically deform, in contrast to the brittle behaviour of bulk sodium chloride crystals at the same temperatures; the nanocrystals have mechanical properties which are a unique feature of their finite size. We also report and compare the global potential energy minima for (NaCl)NCl- using two empirical potentials, and comment on the effect of polarization.Comment: extended version, 13 pages, 8 figures, revte

Comparison of Muscle Transcriptome between Pigs with Divergent Meat Quality Phenotypes Identifies Genes Related to Muscle Metabolism and Structure

Background: Meat quality depends on physiological processes taking place in muscle tissue, which could involve a large pattern of genes associated with both muscle structural and metabolic features. Understanding the biological phenomena underlying muscle phenotype at slaughter is necessary to uncover meat quality development. Therefore, a muscle transcriptome analysis was undertaken to compare gene expression profiles between two highly contrasted pig breeds, Large White (LW) and Basque (B), reared in two different housing systems themselves influencing meat quality. LW is the most predominant breed used in pig industry, which exhibits standard meat quality attributes. B is an indigenous breed with low lean meat and high fat contents, high meat quality characteristics, and is genetically distant from other European pig breeds. Methodology/Principal Findings: Transcriptome analysis undertaken using a custom 15 K microarray, highlighted 1233 genes differentially expressed between breeds (multiple-test adjusted P-value,0.05), out of which 635 were highly expressed in the B and 598 highly expressed in the LW pigs. No difference in gene expression was found between housing systems. Besides, expression level of 12 differentially expressed genes quantified by real-time RT-PCR validated microarray data. Functional annotation clustering emphasized four main clusters associated to transcriptome breed differences: metabolic processes, skeletal muscle structure and organization, extracellular matrix, lysosome, and proteolysis, thereb